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A098453
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Expansion of 1/sqrt(1-4x-12x^2).
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2
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1, 2, 12, 56, 304, 1632, 9024, 50304, 283392, 1607168, 9167872, 52537344, 302239744, 1744412672, 10096263168, 58576306176, 340566147072, 1983765676032, 11574393962496, 67631502065664, 395710949228544
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Central coefficient of (1+2x+4x^2)^n.
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have 2 colors and the U steps can have 4 colors. - Nour-Eddine Fahssi (fahssin(AT)yahoo.fr), Mar 31 2008
a(n) = number of 2 by n matrices with entries in {1,2,3}, same number of 1s in top and bottom rows, and no constant columns. For example, a(1)=2 counts the transposes of (2,3) and (3,2). The number of such matrices with k 1s in each row is binom(n,2k) [choose columns containing 1s] x binom(2k,k) [place 1s in these columns] x 2^n [place 2 or 3 in the topmost available spot in each column and the other of 2,3 in the other spot if not occupied by a 1]. [From David Callan (callan(AT)stat.wisc.edu), Aug 25 2009]
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REFERENCES
| Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
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FORMULA
| G.f.: 1/sqrt((1+2x)(1-6x)); E.g.f.: exp(2x)BesselI(0, 4x); a(n)=sum{k=0..floor(n/2), binomial(n, k)binomial(2(n-k), n)3^k}.
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CROSSREFS
| Cf. A084609, A084770.
Sequence in context: A084128 A044047 A105487 * A180073 A067125 A177782
Adjacent sequences: A098450 A098451 A098452 * A098454 A098455 A098456
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 08 2004
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